nonlinear analysis of a novel three-scroll chaotic system

JAMCJ Appl Math Comput (2012) 39:319–332DOI 10.1007/s12190-011-0523-y

A P P L I E D M AT H E M AT I C S

Nonlinear analysis of a novel three-scroll chaotic system

Xing He · Yonglu Shu · Chuandong Li · Huan Jin

Received: 31 May 2011 / Published online: 18 November 2011© Korean Society for Computational and Applied Mathematics 2011

Abstract This paper reports the nonlinear dynamics of a novel three-scroll chaoticsystem. The local stability of hyperbolic equilibrium and non-hyperbolic equilibriumare investigated by using center manifold theorem. Pitchfork bifurcation, degeneratepitchfork bifurcation and Hopf bifurcation are analyzed when the parameters are var-ied in the space of parameter. For a suitable choice of the parameters, the existenceof singularly degenerate heteroclinic cycles and Hopf bifurcation without parametersare also investigated. Some numerical simulations are given to support the analyticresults.

Keywords Degenerate pitchfork bifurcation · Hopf bifurcation without parameters ·Heteroclinic cycles · Three-scroll chaotic system

Mathematics Subject Classification (2000) 34K-18 · 34C-23 · 34C-37

1 Introduction

Since Lorenz found the first chaotic attractor in a simple mathematic model of aweather system which was made up of three-dimensional autonomous system, moreand more new and interesting chaotic systems were developed, such as the Chen sys-tem [1], the Rössler system [2], the Lü system [3], the Lorenz system family [4], theconjugate Lorenz-type system [5] and so on. These chaotic systems have attractedmore and more attention because chaos has potential application in many fields such

X. He (�) · Y. Shu · C. LiCollege of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. Chinae-mail: [email protected]

H. JinJining Vocational and Technical College, Jining, Shandong, 272037, P.R. China

mailto:[email protected]

320 X. He et al.

as fluid mixing [6], secure communication [7–9], power systems protection [10] andother fields. Recently, chaotic systems have concentrate on theoretically analyzingtheir local and global characteristics to understand chaos better. For example, therehave been some bifurcation analysis concerning the Chen system [11–15], the Lüsystem [16–19] and others. Pitchfork bifurcation, Hopf bifurcation, bautin bifurca-tion, homoclinic bifurcation have been found in these chaotic systems, which showthat these systems have rich nonlinear dynamics.

Recently, Pan et al. [20] presented a new three-dimensional autonomous chaoticsystem with four smooth quadratic terms

⎧⎪⎨

⎪⎩

x = a(y − x) + dxz,

y = (b − a)x − xz + by,

z = −ex2 + xy + cz,

(1)

in which a, b, c, d , e are real parameters. When parameters a = 60, b = 25,c = 5

6 , d = 0.4, e = 0.65, system (1) has the following Lyapunov exponents 0.6971,−0.0392, −3.7549. The system has different fixed points: one saddle point andtwo unstable saddle-focus points. The system is derived from the classic Chen sys-tem, but it can exhibit a three-scroll chaotic attractor as shown in Fig. 1. Henceit has a more complex topological structure than the classical two-scroll attractor.So the system (1) has great potential applications in secure communications be-cause an appropriate chaotic system can be chosen from a catalog of chaotic sys-tem to optimize some desirable factors [28]. Except the dynamic of system (1)in [20], it is interesting to further find out what kind of new dynamics this systemhas.

System (1) was analyzed in [20] from the point of view of its chaotic behav-iors, where Lyapunov exponents, fractal dimension and Poincare map of the newchaotic system are studied, numerically and analytically. The study carried out in thepresent paper may contribute to understand analytically the local stability and bifur-cation. Compared with other chaotic system [11–19], we also analyze the existenceof the degenerate pitchfork bifurcation, singularly degenerate heteroclinic cycles andHopf bifurcation without parameter, and numerically find periodic solution and at-tractors.

Fig. 1 The three-scroll chaoticattractor of system (1) withinitial condition (0.9134,0.6324, 0.0975)

Nonlinear analysis of a novel three-scroll chaotic system 321

This paper is organized as follows. In the next section, the preliminaries relevant tothe bifurcation theory are presented. In Sect. 3, we analyze the stability of hyperbolicequilibrium and non-hyperbolic equilibrium by using center manifold theorem. Thenwe analyze the fold bifurcation and find the versal unfolding of pitchfork bifurcationin Sect. 4. In Sect. 5, the existence of degenerate pitchfork bifurcation and singularlydegenerate heteroclinic cycles are discussed. In Sect. 6, Hopf bifurcation and Hopfbifurcation without parameters are investigated and some numerical examples aregiven to support the analytic results. Finally, some conclusions are stated in Sect. 7.

2 Preliminaries

This section recalls the basic bifurcation theory [21–23].Consider the differential equation

x = f (x,μ), (2)

where x ∈ R3 and μ ∈ R5 are state variables and control parameters, respectively.Assume f is smooth with respect to x and μ0. Suppose that (2) has an equilibriumx = 0 at μ = μ0 and F(x) = f (x,μ0) as

F(x) = Ax + 1

2B(x, x) + 1

6C(x, x, x) + O

(‖ x ‖4),

where A = fx(0,μ0) and, for i = 1,2,3,

B(x, y) =3∑

j,k=1

∂2Fi(ζ )

∂ζj ∂ζk

∣∣∣∣ζ=0

xjyk, C(x, y, z) =3∑

j,k,l=1

∂3Fi(ζ )

∂ζj ∂ζk∂ζl

∣∣∣∣ζ=0

xjykzl.

Definition 1 At the equilibrium x = 0, the bifurcation associated with the appearanceof eigenvalue λ1 = 0 of system (2) is called a fold bifurcation.

Definition 2 At the equilibrium x = 0, the bifurcation associated with the presenceof eigenvalues λ1,2 = ±iω0,ω0 > 0, of system (2) is called a Hopf bifurcation.

3 Equilibrium and local stability

In this section, we apply center manifold theorem [23] to study the local stability ofequilibrium and non-hyperbolic equilibrium. Consider system (1), by simple math-ematical manipulations [20], the system has three equilibria, which are described asfollows

O1(0,0,0), O2

(

−k,−a + ad − db

a + dbk,

2ab − a2

a + db

)

,

O3

(

k,a + ad − db

a + dbk,

2ab − a2

a + db

)

322 X. He et al.

where k =√

2abcd−a2cd

ade+d2eb−ad−ad2+d2b. Obviously, when b = a

2 , O1(0,0,0) is a uniqueequilibrium. If c = 0, the system has a line of equilibria (0,0, z) (z ∈ R). We de-compose the system dynamics (1) to its linear parts A at the origin O1(0,0,0), thebilinear function B(x, y) and the trilinear function C(x, y, z).

A =⎛

⎜⎝

−a a 0

b − a b 0

0 0 c

⎞

⎟⎠ , B(x, y) =

⎛

⎜⎝

d(X1Y3 + X3Y1)

−(X1Y3 + X3Y1)

−2eX1Y1 + X1Y2 + X2Y1

⎞

⎟⎠ ,

C(x, y, z) = 0

where x = (X1, X2, X3)T , y = (Y1, Y2, Y3)

T , and z = (Z1, Z2, Z3)T ∈ R3, and

its characteristic equation at the origin O1(0,0,0) is given by

F(λ) = λ3 + mλ2 + hλ + s = 0, (3)

where m = a − b − c, h = a2 − 2ab − ac + bc, s = ac(2b − a).

Theorem 1 If m > 0,mh − s > 0, s > 0, the equilibrium O1(0,0,0) is asymptoti-cally stable.

Proof According to the Routh-Hurwitz criteria, the real parts of the root λ of (3)are negative if and only if m > 0,mh − s > 0, s > 0. Therefore, the equilibriumO1(0,0,0) is asymptotically stable. �

Theorem 2 If b = a2 , (d+2)(e−1)

c> 0, then the non-hyperbolic equilibrium O1(0,0,0)

is asymptotically stable.

Proof When b = a2 , the origin O1(0,0,0) is non-hyperbolic with three eigenvalues

λ1 = 0, λ2 = c, λ3 = − a2 . Next, we will investigate local stability at the origin by

using the center manifold theorem.In order to put the system (1) in an appropriate form, we make the linear transfor-

mation:⎛

⎜⎝

x

y

z

⎞

⎟⎠ =

⎛

⎜⎝

1 0 2

1 0 1

0 1 0

⎞

⎟⎠

⎛

⎜⎝

x1

y1

z1

⎞

⎟⎠ (4)

bringing system (1) into

⎛

⎜⎝

x1

y1

z1

⎞

⎟⎠ =

⎛

⎜⎝

0 0 0

0 c 0

0 0 − a2

⎞

⎟⎠

⎛

⎜⎝

x1

y1

z1

⎞

⎟⎠ +

⎛

⎜⎝

−(d + 2)(x1 + 2z1)y1

−e(x1 + 2z1)2 + (x1 + z1)(x1 + 2z1)

(d + 1)(x1 + 2z1)y1

⎞

⎟⎠ . (5)

Thus, from the center manifold theory, local stability of O1(0,0,0) can be determinedby studying first-order ordinary differential equations on a center manifold, which can


be represented as a graph over the variable x1 as following:

Wc(0) = {(x1, y1, z1) ∈ R3 | y1 = h1(x1), z1 = h2(x1), |x1| < ε

},

where h1(0) = 0, h1(0) = 0, h′1(0) = 0, h′

2(0) = 0 with ε sufficiently small.Assume that

y1 = h1(x1) = a1x21 + b1x

31 + O

(x4

1

),

(6)z1 = h2(x1) = a2x

21 + b2x

31 + O

(x4

1

).

Thus the center manifold must satisfy

h′1(x1)

(−(d + 2)(x1 + 2z1)y1)

= ch1(x1) − e(x1 + 2z1)2 + (x1 + z1)(x1 + 2z1). (7)

h′2(x1)

(−(d + 2)(x1 + 2z1)y1) = −a

2h2(x1) + (d + 1)(x1 + 2z1)y1. (8)

Substituting (6) into (7) and (8), and we obtain that

a1 = e − 1

c, a2 = 0, b2 = 2a1(d + 1)

a= 2(e − 1)(d + 1)

ac.

Thus one obtains that

y1 = h1(x1) = e − 1

cx2

1 + O(x3

1

),

(9)

z1 = h2(x1) = 2(e − 1)(d + 1)

acx3

1 + O(x4

1

).

Substituting (9) into (5), we obtain the vector field reduced to the center manifold

x1 = − (d + 2)(e − 1)

cx3

1 + O(x4).

So, one can deduce that when (d+2)(e−1)c

> 0, the non-hyperbolic equilibriumO1(0,0,0) is asymptotically stable. �

4 Pitchfork bifurcation

In this section, we apply the projection method [21] and center manifold theorem[23] to study fold bifurcation at the origin for the system (1).

Theorem 3 If b = b0 = a2 , system (1) undergoes a degenerate fold bifurcation at the

origin.

324 X. He et al.

Proof When b = a2 , (3) has a zero root λ1 = 0, this indicates that fold bifurcation

may occur. It is easy to check that the vectors

q =⎛

⎜⎝

1

1

0

⎞

⎟⎠ , p =

⎛

⎜⎝

−1

2

0

⎞

⎟⎠ , (10)

which satisfy

Aq = 0, AT p = 0, 〈p,q〉 =3∑

i=1

piqi = 1,

where 〈·, ·〉 is the standard scalar product in R3.The restriction of system (1) to the one-dimensional center manifold has the form

X = σX2 + O(|X3|), X ∈ R1, (11)

where the coefficient σ can be computed by the formula

σ = 1

2

⟨p,B(q, q)

⟩. (12)

If σ �= 0, and (11) is locally topologically equivalent to the normal form

X = β1 + σX2,

where β1 is the unfolding parameter.But putting (10) into (12) yields: σ = 0. So, fold bifurcation is degenerate. �

From Theorem 3, fold bifurcation is degenerate. In the following, we will dis-cuss pitchfork bifurcation of system (1) and find the versal unfolding of O1(0,0,0)

depending on the original parameters in system (1). We will show that b can be cho-sen as bifurcation parameter and system (1) can exhibit pitchfork bifurcation. Letb = b0 + δ = a

2 + δ, system (1) can be changed into the following system⎧⎪⎨

⎪⎩

x = a(y − x) + dxz,

y = (δ − a2 )x − xz + (δ + a

2 )y,

z = −ex2 + xy + cz,

(13)

where δ is small parameter.We make the linear transformation (4), bringing system (13) into

⎛

⎜⎝

x1

y1

z1

⎞

⎟⎠ =

⎛

⎜⎝

4δ 0 6δ

0 c 0

−2δ 0 −3δ − a2

⎞

⎟⎠

⎛

⎜⎝

x1

y1

z1

⎞

⎟⎠

+⎛

⎜⎝

−(d + 2)(x1 + 2z1)y1

−e(x1 + 2z1)2 + (x1 + z1)(x1 + 2z1)

(d + 1)(x1 + 2z1)y1

⎞

⎟⎠ . (14)


Be similar with the proof of Theorem 2, we make

Wc(0, δ) = {(x1, y1, z1) × δ ∈ R3 × R | y1 = h1(x1, δ),

z1 = h2(x1, δ), |x1| < ε, |δ| < ε},

where h1(0,0) = 0, h1(0,0) = 0, h′1(0,0) = 0, h′

2(0,0) = 0 with ε, ε sufficientlysmall.

And we assume

y1 = h1(x1, δ) = A1x21 + A2x1δ + A3δ

2 + h.o.t.,

z1 = h2(x1, δ) = B1x21 + B2x1δ + B3δ

2 + h.o.t.,

where the high-order terms (h.o.t.) are of the orders O(xk1δ3−k), k = 1,2,3. Be sim-

ilar with (7) and (8), we obtain

A1 = e − 1

c, A2 = A3 = 0, B1 = 0, B2 = −4

a, B3 = 0.

We obtain the vector field reduced to the center manifold

x1 = 4δ

(

1 − 6

aδ

)

x1 − (d + 2)(e − 1)(1 − 8aδ)

cx3

1 + h.o.t. (15)

Finally, rescaling the state x2 =√

| (d+2)(e−1)(1− 8aδ)

c|x1, we get the versal unfolding

x2 = 4δ

(

1 − 6

aδ

)

x2 − x32 + h.o.t. (16)

Thus, the system (1) is strongly topologically equivalent to (16).So, we can get the following theorem.

Theorem 4 If 0 < |b − b0| � 1, (d+2)(e−1)c

�= 0, system (1) undergoes a non-degenerate pitchfork bifurcation at the origin.

Proof For the need of discussing, we rewrite (15) into the following form

x1 = h(x1, δ) = 4δ

(

1 − 6

aδ

)

x1 − (d + 2)(e − 1)(1 − 8aδ)

cx3

1 + h.o.t.

It is easy to check that genericity conditions hold for pitchfork bifurcation [21–23],that is transversality and non-degeneracy conditions

∂3h(x1, δ)

∂x31

= −6(d + 2)(e − 1)(1 − 8aδ)

c�= 0,

(17)∂2h(x1, δ)

∂x1∂δ= 4δ

(

1 − 6

aδ

)

x1 �= 0.

Hence, system (1) undergoes a nondegenerate pitchfork bifurcation. �

326 X. He et al.

5 Degenerate pitchfork bifurcation, singularly degenerate heteroclinic cycles

When the parameter c crosses the zero value, the number of equilibrium will change.If c �= 0, the system has one equilibrium or three equilibria. And the line of equilibriawill exist for c = 0. Compared with pitchfork bifurcation, the difference is that, thenumber of equilibrium is infinite at the critical value. We have called the bifurcationa degenerate pitchfork bifurcation, due to the line of equilibria which exist for c = 0.This phenomenon has already been observed in the differential equation [24, 25].

For c = 0, system (1) becomes⎧⎪⎨

⎪⎩

x = a(y − x) + dxz,

y = (b − a)x − xz + by,

z = −ex2 + xy,

(18)

whose equilibria are Oz(0,0, z), The characteristic polynomial at Oz(0,0, z) is givenby

p(λ) = λ(λ2 + τλ + W

) = 0.

Where τ = a − b − dz, W = −2ab + bdz + a2 + az and � = τ 2 − 4W = d2z2 −2z((a −b)d +2bd +2a)+ (a −b)2 +8ab−4a2. Therefore the eigenvalues are givenby

λ1 = 0, λ2,3 = −τ ± √�

2

with the corresponding eigenvectors

v1 = (0,0,1), v2 =(

a + b − dz + √�

2a,1,0

)

,

v3 =(

1,−a + b − dz + √�

2(b − a − z),0

)

.

If W < 0 and � > 0, the eigenvalues λ2,3 are real with opposite signs. Then takinginto account the eigenvector v2,3, the system has a normally hyperbolic saddle at theequilibrium point Oz(0,0, z).

If W > 0 and � > 0, the eigenvalues λ2,3 are real with same signs. Then takinginto account the eigenvector v2,3, the system has a normally hyperbolic stable orunstable node at the equilibrium point Oz(0,0, z).

If � < 0, the eigenvalues λ2,3 are complex conjugate. Then taking into accountthe eigenvector Im(v2) and Re(v2), the system has a normally hyperbolic stable focusif τ > 0 and unstable focus if τ < 0 at the equilibrium point Oz(0,0, z).

If W = 0, the equilibrium point Oz(0,0, z) is more degenerated, having two van-ishing eigenvalues.

By above analysis, the system presents an infinite set of singularly degenerate het-eroclinic cycles in system (1) when a > 0, b > 0, c = 0, d > 0, e > 0. These cycles


Fig. 2 Singularly degenerateheteroclinic cycles of system(18) with initial valueOz(0.001,0.001,−30)

consist of invariant sets formed by a line of equilibrium points together with hetero-clinic cycles connecting two of the equilibria. Here each one of these cycles is formedby one of the one-dimensional unstable manifolds of the normally hyperbolic saddleOz(0,0, z), which connects Oz with and the normally hyperbolic focus Oz1(0,0, z1).For more details, see [26, 27]. When a = 60, b = 25, c = 0, d = 0.4, e = 0.65, basedon numerical investigation, system (1) has infinitely many degenerate heteroclinic cy-cles, each one of these cycles is formed by one of the one-dimensional unstable man-ifolds of the normally hyperbolic saddle Oz(0,0, z), z < − 60

7 , which connects Oz

with and the normally hyperbolic focus Oz1(0,0, z1), −3.8074 < z1 < 1928.8074.Let the normally hyperbolic saddle be Oz(0,0,−30), singularly degenerate hetero-clinic cycles is created in Fig. 2.

6 Hopf bifurcation

6.1 Generic Hopf bifurcation

In this section we apply higher dimension Hopf bifurcation theory and sym-bolic computations to perform the analysis of parametric variations with respectto dynamical bifurcation. For simplicity, we fix a = d , e = 1, b = a−1

3 , un-der these parameters conditions, the equilibrium of system (1) is O1(0,0,0),O2(−√

c,−2√

c,−1), O3(√

c,2√

c,−1). Because the system is invariant under thetransformation (x, y, z) → (−x,−y, z), we only consider Hopf bifurcation of thesystem (1) at O3(

√c,2

√c,−1). Under the following linear transformation

⎧⎪⎨

⎪⎩

x1 = x − √c,

y1 = y − 2√

c,

z1 = z + 1,

328 X. He et al.

which transform the equilibrium O3(√

c,2√

c,−1) to the origin, system (1) become⎧⎪⎨

⎪⎩

x1 = −2ax1 + ay1 + a√

cz1 + ax1z1,

y1 = 2−2a3 x1 + a−1

3 y1 − √cz1 − x1z1,

z1 = √cy1 + cz1 − x2

1 + x1y1.

(19)

Linearizing system (19) at the origin now yields Jacobian matrix A =( −2a a a

√c

2−2a3

a−13 −√

c

0√

c c

)

.

And its characteristic equation is given by

F(λ) = λ3 +(

1

3− c + 5

3a

)

λ2 +(

2c

3− 5ac

3

)

λ + 2

3a2c + 4

3ac = 0. (20)

Suppose (20) has a pair of imaginary roots λ2,3 = ±ω0i, ω0 > 0. Substituting it into(20) yields:

−ω30i −

(1

3− c + 5

3a

)

ω20 +

(2c

3− 5ac

3

)

ω0i + 2

3a2c + 4

3ac = 0,

ω30i −

(1

3− c + 5

3a

)

ω20 −

(2c

3− 5ac

3

)

ω0i + 2

3a2c + 4

3ac = 0.

By calculation, we get(

1

3− c + 5

3a

)

ω20 = 2

3a2c + 4

3ac,ω2

0 =(

2c

3− 5ac

3

)

.

From (20), one obtains

λ1 = −(

1

3− c + 5

3a

)

, λ2,3 = ±iω0.

Thus, when ( 13 − c + 5

3a) > 0, ω20 > 0. We can select c = c0 = 31a2+7a−2

15a−6 and

a ∈ (−7−3√

3362 , −7+3

√33

62 ). Hence, Hopf bifurcation can appear at the point O1.

Theorem 5 If c = c0 = 31a2+7a−215a−6 and a ∈ (−7−3

√33

62 , 13−√817

162 ) ∪ ( 13−√817

162 ,−7+3

√33

62 ), (20) has three eigenvalues: the first is negative, the other two are a pairof purely imaginary conjugate roots, system (1) undergoes a Hopf bifurcation at theorigin.

Proof When c = c0 and a ∈ (−7−3√

3362 , −7+3

√33

62 ), (20) can be changed into

f (λ) =(

λ + 1

3− c + 5

3a

)(λ2 + ω2

0

) = 0.

So (20) has a pair of purely imaginary conjugate roots λ2,3 = ±iω0 =±i

√2c03 − 5ac0

3 and a negative root λ1 = −( 13 − c0 + 5

3a).


Fig. 3 The phase diagram ofsystem (1) with initial valuex0 = (0.5,0.9,−1)

Suppose the roots of (20) are λ(β) = μ(β) + iγ (β).

From (20), we have

λ′(c) = − −λ2 + ( 23 − 5

3a)λ + 23a2 + 4

3a

3λ2 + 2λ( 13 − c + 5

3a) + 2c3 − 5ac

3

.

So,

μ′(c0) = Re(λ′(c0))|λ=iω0 = −81a2 + 13a + 2

18(ω20 + ( 1

3 − c0 + 53a)2)

�= 0.

So, when c = c0, system (1) undergoes a Hopf bifurcation at the equilibrium. �

Fix a = 0.1, when c = c0. The phase diagrams of system (1) is shown in Fig. 3.

6.2 Hopf bifurcation from lines of equilibria without parameters

In this section, we will numerically investigate in the global consequences of thechanges in the local stability of the equilibrium point Oz(0,0, z). From part 5, when� = τ 2 − 4W = d2z2 − 2z((a − b)d + 2bd + 2a) + (a − b)2 + 8ab − 4a2 < 0, theequilibrium point Oz(0,0, z) is normally hyperbolic stable focus if τ > 0 and unsta-ble focus if τ < 0. Fix parameters a = 60, b = 25, c = 0, d = 0.4, e = 0.65. Hence,the equilibrium point Oz(0,0, z) has complex conjugate eigenvalues with negativereal part for z ∈ (−3.8074,87.5) and positive real part for z ∈ (87.5,1928.8074). Sothere is a change in the local stability of the equilibrium as z crosses z0 = 87.5.

When z < 87.5, the equilibrium point Oz(0,0, z) is normally hyperbolic stablefocus, the phase diagram of system (18) is shown in Figs. 4 and 5.

Since z = 87.5, the equilibrium point Oz(0,0, z) is a vague attractor like a cylin-der. The cylinder attractor is shown in Fig. 6.

When z > 87.5, the equilibrium point Oz(0,0, z) is normally hyperbolic unstablefocus. Figure 7 shows the phase diagram of system (18).

330 X. He et al.

Fig. 4 The phase diagram ofsystem (18) with initial valueOz = (0.001,0.001,85).Integration time: [0.1,10]

Fig. 5 The phase diagram ofsystem (18) with initial valueOz = (0.001,0.001,85).Integration time: [5,10]

Fig. 6 The cylinder attractor ofsystem (18) with initial valueOz = (0.001,0.001,87.5).Integration time: [5,10]

The numerical analysis suggests that system (18) has an attractor which are sym-metric in relation to the Oz(0,0, z) when z ∈ (87.5,120). This attractor is given byan invariant surface filled with normal hyperbolic stable periodic orbits. A sketch ofsuch a surface is shown in Fig. 8.


Fig. 7 The phase diagram ofsystem (18) with initial valueOz = (0.001,0.001,90).Integration time: [0.2,10]

Fig. 8 The attractor of system(18) with initial valueOz = (0.001,0.001,100).Integration time: [0.2,25]

7 Conclusion

In this paper, the pitchfork bifurcation and Hopf bifurcation have been investigatedfor the system (1) by using the higher dimensional bifurcation theory and symboliccomputation, and the existence of bifurcation theorem has also been theoreticallyproved under certain conditions. Moreover, we have discussed Hopf bifurcation with-out parameter and singularly degenerate heteroclinic cycles for a suitable choice ofthe parameter.

Acknowledgements This research is supported by the National Natural Science Foundation of ChinaGrant No. 11171360, No. 60974020 and the Fundamental Research Funds for the Central Universities ofChina (Project No. CDJZR10 18 55 01).

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